Problem 2. Suppose X1, ..., In are samples from a distribution with mean u and variance o. We consider a class of estimator of u using the weighted average of the samples P ( c1, C2, . . ., Cn) = ciXi. i= 1 where ci are constants and sum to 1 Eci = 1. i= 1 The sample mean X is f(?, . . ., ?). (a). Show that u(C1, C2, . .., (n) is an unbiased estimator of u. (b). Compute the following ratio R = MSE((c1, C2, . . ., Cn)) MSE(X) (c). From (b) we see R only depends on c1, C2, . .., Cn. Consider R as a function of (C1, C2, . .., Cn) and find the choice of (c1, C2, . .., Cn) that minimizes R. What is your conclusion? [hint: you may want to use the following result (see handout 1, page 9) I : ) 2 n - 152 > 0. n thus for our problem, take ci = Ci I'M Ci) 2 n n n2